A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Wednesday, January 15, 2014

Halfway through our winter break I paused in my course
preparations, panicking that I'd spent nearly all of my time planning my
HON 479 course and had done nearly nothing to prep for Linear Algebra II.
Though I had a rough framework for the course's structure (semiregular
homework assignments, a few take-home exams, student-led projects,
presentations, and discussions), I had almost no idea what content I would include in the course.

After a few moments (okay, maybe a few hours), the panic passed. I realized the futility of overplanning, a futility reconfirmed by the survey of my Linear II students' background I performed on Monday. The 23 students in that class come to me having taken Linear Algebra I from no fewer than five different faculty members in my department, as long ago as two and a half years back. These faculty include me and one of my colleagues who shares my penchant for student-centered, application-based teaching, a couple folks who typically offer a blend of applications and theory (one with a much more student-centered approach than the other), and a fifth who focuses exclusively on abstraction and theory and whose teaching style can only be described as "traditional." Needless to say, my 23 students come to me with extremely diverse linear algebraic backgrounds. It's unlikely that, beyond a few basic principles (row reduction, linear (in)dependence, bases, determinants, eigenvalues and -vectors, etc.) they all will have studied, they'll have any content knowledge in common. In the end there's really very little I can do to accommodate them all: no matter what static plan for the course that I could come up with, it would no doubt lose some and bore most of the others.

This realization was liberating. Instead of putting forth a particular course of study, I could let the students take the lead, offering them the chance to investigate topics in which they are interested, sharing their investigations with each other in the form of in-class presentations, discussions, and problem sets. I'm going to ask every student to take a turn, working with one or two of her or his peers, leading the class in the study of a topic of her or his choosing. For those who might not know what direction they'd like to head in, I made a list of potential topics, many of which likely made an appearance in some students' first-semester Linear I courses:

To help everyone get to the point where we can approach some of these topics, I'm spending the first week or two on review, wherein the students are taking turns, in groups of three or four, presenting on the various "basic principles" I listed above. It's going well so far. "Is this useful at all?" I asked after a couple of presentations this morning. "Should we keep doing this?" There was almost unanimous agreement that yes, we should. So we'll keep it up.

Tuesday, January 14, 2014

The Honors Program has become a victim of its own success, in a way: so many students are now pursuing Distinction as a University Scholar that I'm finding it necessary to offer not one but two sections of HON 479 this term...and perhaps in (nearly?) every term for the indefinite future. Of course, by the time that I realized the need for a second section of the course (about halfway through last semester's advising period), it was really too late to find another instructor to teach that section, and besides, I'd prefer to have a single instructor for both sections, for consistency's sake. Of course, that meant that the instructor for the first section would also be the instructor for the second section, even if it meant (as it did) teaching a course over the normal load. Of course, that instructor is me.

Genius that I am, I scheduled the two sections to meet back-to-back, 100 minutes apiece with only ten minutes in between, every Tuesday and Thursday. Today was our first meeting. By the end of the first section my throat hurt, and by the end of the second I had nearly no voice: there are so many moving parts to this class that I've just got to spend much of the first class meeting pointing out just how all of those parts fit together and more in a meaningful way.

Throw in a minor student medical emergency, a pressing tech issue facing the school's student-run TEDx chapter (for which I'm the faculty adviser), various administrivia and bureaucratic bullshit, and a two-hour sojourn in Asheville Catholic School's gymnasium, where I helped a friend out as a middle-school science fair judge, and you've got a hell of a day. I'm sore-throated and brain-dead, and I'm tired as hell.

But I'm happy. I've got high hopes for this term. I feel like last semester gave me a good grip on 479, and I had a fantastic first meeting of Linear Algebra II yesterday (the 23 students in that class had five different instructors for Linear I!), a course which I'll be teaching from a nearly total project-based perspective.